Conversely, given a hyperbolic metric \(h\)  on the disk \({\mathbb {D}}\)  such that the identity between \({\mathbb {D}}\)  (equipped with its standard conformal class) and \(h\)  is harmonic, there is a unique hyperbolic metric \(h^{\prime }\)  on \({\mathbb {D}}\)  (obtained by integrating the Schwarzian equation, see [1]}) such that the identity map from \({\mathbb {D}}\)  to \(({\mathbb {D}},h^{\prime })\)  is harmonic, with Hopf differential opposite to that of the harmonic map from \({\mathbb {D}}\)  to \(({\mathbb {D}}, h)\) . The identity map from \(({\mathbb {D}},h)\)  to \(({\mathbb {D}},h^{\prime })\)  is then minimal Lagrangian. Given \(K\in (-1,0)\) , the construction of the previous section then associates to the pair \((h,h^{\prime })\)  an immersion from \({\mathbb {D}}\)  to \({\mathbb {H}}^3\)  with induced metric \((1/|K|)h\)  and third fundamental form \((1/|K^*|)h^{\prime }\) , and by construction the identity is harmonic from \(({\mathbb {D}}, (1+K)I+I\hspace{-2.84544pt}I\hspace{-2.84544pt}I)\) .
